A multiphase model for the flow of foaming heavy crude oil in a porous medium, with illustrative calculations

This report details work done by the author during a six month secondment from Schlumberger Cambridge Research to Intevep during April to October 1997. A mathematical description for the flow within a porous permeable medium of oil containing dissolved gas, and capable of liberating that gas when the pressure is dropped to form small bubbles or regions of connected gas, is presented. The intention is that the model will serve as a description for the flow within the reservoir rock of a foamy heavy crude oil. The model consists of a set of conservation or transport equations, together with constitutive functions describing mass transfer processes. Special emphasis is put on investigating the consequences of finite process rates, and non-linearity, in these mass transfer models. The description draws on earlier work of [Joseph], but differs from that in that relative motion between liquid and gas bubbles is allowed, a more elaborate description of exchanges between the phases is used, and variables describing some aspects of the microscale state of the gas bubbles are tracked in time. The model is used to calculate pressure and volume fraction distributions and time histories in three different, experimentally realisable, situations: slow depressurization of a fluid sample within a sand pack; steady flow through a core under conditions where gas bubbles are evolved and grow during flow within the core; unsteady and non-uniform flow from an initially pressurised and fluid saturated core when the pressure is suddenly dropped at one end. In the slow depressurization simulations it is found that a mass transfer model based on nucleation theory is capable of reproducing the non-monotonic pressure-time histories observed in experiments, whereas a simpler linear kinetics model is not. In the core flow simulations, again with the nucleation model, a flow rate threshold is found, below which bubble nucleation occurs in a narrow localised zone at pressures close to but a finite amount below the bubble point, and above which nucleation occurs everywhere within the core, continuing at pressures very far below the bubble point. Overall, the size of bubbles is found to increase as the flow rate through the core decreases. It is conjectured that these behaviours may play some part in causing, directly or as a precursor, the experimentally observed dependence of critical gas saturation on flow rate. The approximate core depressurization simulations illustrate some general features of the pressure and volume fraction distributions in this situation, and are of value for checking the results of numerical computations in which all the non-linear terms in the governing equations are present. This version printed: 06/19/98 1:04 PM